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CHEM 332L

Physical Chemistry Laboratory

Revision 2.1

Conjugated Bonding in Cyanine Dyes: A "Particle In A Box" Model

In this laboratory exercise we will examine the Visible Spectra of a series of Cyanine Dye

molecules and determine max for each compound. These results will then be compared with

max values obtained by treating the conjugated electrons of the molecules as a free-electron

gas confined to a one-dimensional box whose length is that of the molecule.

The "Particle In A Box" problem is one that is ubiquitous in elementary courses on quantum

mechanics. This is because it illustrates a number of key principles concerning quantum

mechanics and because the solution of the Schrodinger Wave Equation for this system isanalytically solvable. In this model system a "particle" of mass m is confined to a one-

dimensional "box" along thex-axis betweenx=0 andx=L. Within the "box" the particleexperiences no potential, outside the "box" it is subject to an infinite potential. This model

results in a Wave Equation ( = E) within the "box" of:

(Eq. 1)

Solution of this 2nd

Order Linear Homogeneous Differential Equation, after application of

appropriate boundary conditions and normalization, yields the following wave functions:

n= 1,2,3, 0 x L (Eq. 2)

with associated energies:

(Eq. 3)

Absorption of a photon of appropriate energy:

E = Eph = h = hc/ (Eq. 4)

where and are the wavelength and frequency of the photon respectively, will lead to an

energy transition as pictured below.


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Here, nu and nlare quantum numbers associated with the upper and lower quantum states

involved in the transition, respectively. For a Particle In A Box, this transition involves an

energy change of:

E =

(Eq. 5)

Although seemingly abstract, Hans Kuhn has shown (J. Chem. Physics, 1949) that the Particle In

A Box model can be used to predict the wavelength of maximum absorbance (max) for a series

of organic dyes possessing highly conjugated bonds and James R. Arnold has used this model to

predict the energies of single covalent bonds (J. Chem. Physics, 1956). In these constructionsthe molecular "bond" is modeled as a one dimensional box containing a free electron gas

consisting of the bonding electrons. At least qualitatively, both Kuhn and Arnold obtain results

which are correct.

We will follow Kuhn's lead and predict max for a series of Cyanine dyes, which will then becompared with experimental determinations to judge the quality of the predictions.

Consider the dye 1,1'-diethyl-4,4'-cyanine iodide:

The conjugated series of double bonds 1-4 extends from the Nitrogen atom on one end of themolecule to the Nitrogen atom on the opposite end. TheNelectrons (10 e

-'s in this case; the

lone pair on the Nitrogen must be included in the count):

N = 2j + 2 wherej = # of double bonds (Eq. 6)

involved in this conjugated system can be thought of as occupying molecular electronic states

according to the Pauli Exclusion Principle.


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An electronic transition can occur from the Highest Occupied Molecular Orbital (HOMO) to the

Lowest Unoccupied Molecular Orbital (LUMO) when a photon is absorbed according toEquation 4. For our dye molecules in solution, we will assume:

max = hc / E (Eq. 7)

The HOMO-LUMO transition will then occur between theN/2 and theN/2+1 levels. Thus, we

can estimate E according to Equation 5 as:

E = EN/2+1 - EN/2

=

= (Eq. 8)

This result can be used to predict the max for the compound's spectrum, if we knowL, the length

of the molecular "box."

To a first approximation, theseNelectrons can be modeled as confined to a one-dimensional box

of length equal to that of the conjugated chain, designatedL for our purposes. Based on thelength of the C=C bonds involved in the conjugated chain,L can be determined according to:

L = Nx l where l= length of C=C bond = 139 pm (Eq. 9)

According to Kuhn, the length of the conjugated chain should be taken as "the length of the

polymethine ziz-zag chain between the nitrogen atoms plus one bond distance to either side."

The purpose of including a "bond distance" on either side of the Nitrogen atoms is to include the

"distance" occupied by the lone pairs of the Nitrogen. Further, the aromatic rings at the ends ofthe conjugated chain provide a soft rather than abrupt rise in the potential energy at the ends of

the "box." This softness can be accommodated using an appropriately chosen empirical fittingparameter. We are only interested in the qualitative alignment of the model with the

experimental data, so we will not account for these details.

Based on the Particle In A Box model, we expect the electronic transition to produce a fairlysharp spectral band. However, for solutions of these compounds, the spectral band is observed to

be fairly broad; on the order of tens of nanometers.


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This can be accounted for by considering molecular vibrations within the transitioning

molecules. If the electronic transitions from state to state occur within a time interval muchshorter than that of a molecular vibration, the absorption from many molecules shows up as a

broad band; those molecules that have their conjugated bond system stretched or compressed

during a transition, absorb at longer and shorter wavelengths than the absorption maximum, max.

So, in this study, we will measure the max for a series of cyanine dyes and compare the resultswith those expected based on the Particle In A Box model. These results will then provide us

with some insight into the electronic structure of these compounds and the nature of a chemical

bond.


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Procedure

Prepare 10-4

M Methanol solutions of the following three cyanine dyes:

Take the Visible Spectrum of each solution and determine max (include an error estimate) foreach compound. If the absorbance maximum is much greater than unity, your solutions should

be diluted and the spectrum re-taken. This will minimize the occurrence of dimers which could

affect the determination of the max.


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Data Analysis

1. Determine max for each compound whose spectrum was taken, using the Particle In A Boxmodel. Compare each result with the experimental value. Comment.

2. Compare your experimentally determined max values with those of the literature.Comment.

3. We wish to examine James R. Arnold's approach to understanding covalent chemical

bonding between two atoms, A and B. Using this approach, each atom is modeled as a onedimensional Particle In A Box. For atom A the "box" will have a width of 2rA, where rA is

the atomic radius. Similarly for atom B, a "box" of width 2rB is used. In each case, the

particle is a free electron. Once atoms A and B bond to form the molecule A-B, a "box"

containing two electrons of length rA + rB + R, where R is the bond distance, is used. Thisis as pictured:

The bond energy can be approximated by the difference between the energy of themolecule and that of the two atoms, assuming the each system is in the ground state (n =

1):


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Bond Energy = EA + EB - EA-B (Eq. 10)

Each energy Ei can be determined via Equation 3. (Recall EA-B involves two electrons in

the ground state.)

a) Estimate the S-S bond energy using Arnold's approach as outlined above. According toHuheey, the van der Waals radius (rvdw) of a Sulfur atom is 1.80 and the Sulfur-Sulfurbond length in S8 is 2.05. Experimentally this bond energy has been determined as49 kcal/mol. Calculate the percentage error in your estimate.

b) Arnold sets the atomic radii ri as being proportional to an atom's non-bonding radius:

ri = krvdw

where kis an empirical fitting parameter. Arnold finds a value ofk= 1.185 provides a

"best fit to all the data." Use this modified approach to determine the S-S bond energy.Again, calculate the percentage error in this estimate.

c) Assuming this model is a reasonable picture of reality, what is the physical source of

the bonding between the two atoms A and B?

d) In a real bond, what is the physical source of the bonding between the two atoms A and

B?


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References

Arnold, James R. "'Smoothed Potential' Theory of Chemical Binding" J. Chem. Physics24(1956) 181.

Brooker, L.G.S., Keyes, G.H. and Williams, W.W. "Color and Constitution. V. The Absoprtionof Unsymmetrical Cyanines. Resonance as a Basis for a Classification of Dyes" J. Am.

Chem. Soc.64 (1942) 199.

Fisher, Nellie I. and Hamer, Frances M. "A Comparison of the Absorption Spectra of SomeTypical Cyanine Dyes" Proc. Roy. Soc., Ser. A 154 (1936) 703.

Gerkin, Roger E. "A Molecular Spectral Corroboration of Elementary Operator Quantum

Mechanics" J. Chem. Ed.42 (1965) 490.

Goodisman, J. Contemporary Quantum Chemistry: An Introduction Plenum Publishing, New

York, 1979.

Huheey, James E., Keiter, Ellen A. and Keiter, Richard L. Inorganic Chemistry: Principles of

Structure and Reactivity, 4th

Ed. Harper Collins College Publishers, New York, 1993.

Kuhn, Hans "A Quantum-Mechanical Theory of Light Absorption of Organic Dyes and Similar

Compounds" J. Chem. Physics17 (1949) 1198.

Sime, Rodney J.Physical Chemistry: Methods, Techniques, and Experiments Saunders College

Publishing, Philadelphia, 1990.

Shoemaker, David P.; Garland, Carl W. and Nibler, Joseph W. Experiments in PhysicalChemistry McGraw-Hill, New York, 1996.

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